Labor Market Cycles and Unemployment Insurance Eligibility

Transcription

1 Labor Market Cycles and Unemployment Insurance Eligibility Miquel Faig Min Zhang y Febrary 16, 2008 Abstract If entitlement to UI bene ts must be earned with employment, generous UI is an additional bene t to an employment relationship, so it promotes job creation. If individuals are risk neutral, UI is fairly priced, and the UI system prevents moralhazard unemployed workers, the generosity of UI has no e ect on unemployment. As with Ricardian Equivalence, this result should be useful to pinpoint the e ects of UI to violation of its premises. In itself, the endogenous entitlement of UI bene ts does not resolve if the Mortensen-Pissarides model is able to generate realistic cycles. However, it brings some insights into this debate: The widespread concern in the design of UI systems to minimize moral-hazard unemployment only makes sense if workers have su ciently high values of leisure (80 percent of labor productivity in our baseline calculation for the United States). The fact that the generosity of UI has potentially a small e ect on unemployment reconciles a high response of unemployment to changes in labor productivity with a small response to changes in UI bene ts. JEL classi cation: E24 E32 J64 Keywords: Search, Matching, UI Eligibility, Business Cycles, Labor Markets Mailing Address: University of Toronto, Department of Economics, Institute for Policy Analysis, 140 St. George Street, room mfaig@chass.utoronto.ca. y Mailing Address: Sidney Smith Hall, University of Toronto, 100 St. George St., Rm 4072, Toronto, Ontario M5S 3G3, Canada. maggie.zhang@utoronto.ca 1

2 1 Introduction Most models of employment ows in the labor market assume that workers automatically qualify for unemployment insurance (UI) bene ts while they are searching for a job. As pointed out by Mortensen (1977), Burdett (1979), and Hamermesh (1979), this simplistic view of how a UI system operates may lead to highly misleading conclusions about its impact on the labor market. To avoid this criticism, several papers taking into account more realistic features of the UI systems have emerged. However, because of the institutional complexities of actual UI systems, these models rely exclusively on numerical methods for their analyses, and, they either assume an exogenous distribution of real wages (Andolfatto and Gomme, 1996) or a non-standard mechanism for its determination (Brown and Ferrall, 2003). In this paper, we advance an analytically tractable version of the standard Mortensen-Pissarides search and matching model in which workers are not always entitled to UI bene ts because such an entitlement must be earned with prior and not too distant employment, and it can be lost if workers quit their jobs voluntarily or refuse job o ers. If UI bene ts are unconditionally received while searching for a job, they represent an opportunity cost of employment, and improve the bargaining position of workers while negotiating over wages with their employers. As a result, UI bene ts reduce the expected pro ts of lling a vacancy, and hurt rms incentives for job creation and therefore employment. In contrast, if UI bene ts are conditional on prior employment, they are no longer an opportunity cost, but an indirect bene t of employment. True enough, once workers become eligible for UI their bargaining position improves and so their salaries rise. However, this is well anticipated by all involved, so UI bene ts reinforce the bargaining position of rms dealing with workers who are not yet eligible for UI. Consequently, UI bene ts promote the value of lling a vacancy and stimulate job creation. This is the entitlement e ect stressed by Mortensen (1977), Burdett (1979), and Hamermesh (1979) but operating through a new channel. In those papers, the desire to earn UI entitlement reduces the reservation wages of workers searching for jobs, which in turn reduces unemployment. In our model, the entitlement e ect operates through the bargaining positions 2

3 of rms and workers. The UI bene ts, making the employment match more attractive to workers, enable rms to appropriate a larger fraction of the match surplus, which translates into a stronger incentive to post vacancies. Even if generous UI bene ts encourage job creation, they may hurt employment when we take into account the nancial costs of the UI system. When the UI program is funded by the UI contribution fees paid by employed workers, a generous UI system is also an expensive one, and the large fees needed to maintain it lower the workers desire of being employed, and so the value of lling a vacancy. Therefore, an expensive UI system imposes a downward pressure on employment. Based on these two competing e ects of UI bene ts, we obtain the following analog to Ricardian Equivalence: If UI rules can prevent the moral hazard behavior of becoming or remaining unemployed, each employed worker is charged a fair unemployment insurance fee, and utilities are linear, then the generosity of UI bene ts, the duration of these bene ts, and the time it takes to become eligible for UI are all irrelevant to the determination of output, vacancies, and unemployment. Like Ricardian Equivalence, this irrelevance result should be a useful benchmark to pinpoint the economic e ects of a UI system as violations of its premises. That is, the economic relevance of a UI system must be found on the risk aversion of workers, the "unfair" pricing of UI services, and moral hazard. If workers are risk averse, UI provides the valuable service of smoothing consumption uctuations in the presence of employment shocks. The Mortensen-Pissarides model typically abstracts from this purpose by assuming linear utilities, and we follow this tradition in this paper. If UI contributions, or equivalently taxes that ultimately fall on employed workers, do not match the expected present discounted value of the UI bene ts to be received during unemployment spells, the entitlement e ect of UI bene ts does not o set the opportunity cost of nancing these bene ts. So, the UI system may either increase or decrease employment depending on if the insured workers in question are subsidized or not from other sources of government revenue. Finally, workers may alter the hazard of being or remaining unemployed by changing their search intensity, refusing job o ers, or strategically quitting jobs once 3

4 they are eligible for UI. In a balance between tractability and realism, the present contribution focuses on the third of these hazard; that is, the assumptions of the model allow for moral hazard quits, but abstract from the e ect of UI on search intensity and the acceptance of job o ers. The possibility of moral hazard quits opens the door to an interesting form of multiple equilibria. For a generic set of parameters values, two types of equilibria coexist: A "good" equilibrium where workers do not quit once they are eligible for UI and a "bad" equilibrium where such quits occur. In the good equilibrium, few workers collect UI bene ts and many are employed, so the UI contributions required to nance the UI program can be low, which in turn makes it undesirable to quit a job to collect UI bene ts. In the bad equilibrium, many workers collect UI and few contribute to the UI system. Hence, UI contributions need to be high, which induces workers to quit as soon as they can collect UI. This multiplicity of equilibria is a reminder that fully rating UI contributions is not enough to curtail the moral hazards induced by a UI system. When our model is confronted with data from Canada and the United States, it o ers the following insights on the current debate about the appropriateness of the Mortensen- Pissarides model in explaining the cyclical uctuations in the labor market. First, the eligibility rules of these two countries show a major concern to avoid moral hazard quits and such concern is only meaningful if the value of leisure is not too low. For example, in our baseline calibration with United States data, we calculate that for values of leisure below 80 percent of labor productivity, workers would never quit to collect UI even if they knew they would be able to collect the statutory UI replacement rate (40 percent) with probability one. Hence, even if the obvious political and social concern about unemployment implies that the value of leisure cannot be close to labor productivity (see Mortensen and Nagypál, 2007), the further concern to avoid moral hazard unemployment implies that the value of leisure cannot be too low either. Second, our calibrations of the model to cyclical data from Canada and the United States require similar values of leisure as a percentage of labor productivity (around 54 percent) even if the levels of generosity of UI systems in these two countries are quite di erent. Third, our calibrations are able 4

5 to generate realistically large labor market cycles in response to productivity shocks, even if unemployment responds little to correlated changes in taxes and UI bene ts. The rest of the paper is organized as follows. Section 2 describes our stochastic version of the Mortensen-Pissarides model with a UI system in which individuals need to earn their UI eligibility. Section 3 analyzes a deterministic version of the model. Section 4 ts the model to data on the labor market cycles in Canada and the United States. Finally, section 5 concludes. 2 The Model Our model is a stochastic version of Pissarides (1985) search model. To simplify algebraic expressions, we use continuous time in the analysis of this and the following section, although a discrete time version of the same model will be employed in the numerical simulations of Section Basic Environment In the economy, there is a continuum of measure one of workers, and a large measure of potential rms with free entry into the labor market. Both workers and rms are in nitely lived, risk neutral, maximize their expected utilities, and discount future utility ows at the common rate r: Production requires the cooperation of one worker and one rm. For this cooperation to take place, workers and rms must rst enter the labor market and search for a suitable partner. Once a match has been formed, it produces a ow of output p until it breaks down. The productivity p; common to all matches in the economy, follows a Markov jump stochastic process with a constant arrival rate and takes values in a nite support P 2 R+: n The surplus from a match is split between the two parties according to a generalized axioms of Nash. Finally, employment matches dissolve either exogenously as a result of separations which come at an arrival rate s; or endogenously if breaking the match is in the interest of one of the two parties. The key feature we introduce to this standard environment is that workers do not 5

6 always collect unemployment insurance bene ts (UI) while they are searching for jobs. For workers to be eligible for UI, they must rst be employed for a while, and bene ts do not last forever. Furthermore, UI bene ts are meant to be collected for workers who lose their jobs involuntarily, although, to be realistic, we allow some workers who quit to successfully pretend to have lost their jobs involuntarily. To capture these features in a tractable way, the following assumptions are made. Newly employed workers are not eligible for UI, and eligibility is the outcome of a jump stochastic process with an arrival rate g: Eligible workers always collect UI if they su er an exogenous separation from their jobs, but if they quit, they collect UI with probability 2 [0; 1]. Finally, unemployed workers collecting UI lose eligibility either when they are o ered a job or as a result of a jump stochastic process with an arrival rate d: Unemployment insurance is provided by a government, which nances the UI system with a mandatory state dependent contribution fee p collected from all employed workers. Since the government can borrow and save at the interest rate r; the UI program can run de cits or surpluses over time. Later on, we will allow for permanent de cits or surpluses by introducing general taxation and a public good. All workers are identical in terms of preferences and abilities, and supply labor inelastically. The only di erence across workers lies in the UI eligibility, which is indicated by the individual state variable i : 8 >< 1; if the worker is eligible for UI, and i = >: 0; otherwise. Net of the UI contribution fee, employed workers earn a state dependent wage rate w i p, where the superscript i denotes the UI eligibility state, and the subscript p denotes the productivity state. The wage rate w i p depends on UI eligibility because UI bene ts raise the opportunity cost of employment, so they improve the worker s bargaining position in the negotiations to split the match surplus. Unemployed workers receive a ow utility from leisure `; and, if eligible for UI, they also receive UI bene ts b. To avoid uninteresting possibilities, both ` and b are assumed to be positive, and ` is assumed smaller than the 6

7 production in a match net of the UI contribution fee: ` < p p for all p 2 P: However, the total opportunity cost of employment for a worker who is entitled to collect UI, ` + b; may surpass production net of the UI fee for some realizations of p, which raises the possibility of moral hazard quits. All rms possess the same production technology and preferences. Each one of them chooses to either stay idle or be active in the labor market. An active rm searching for a worker posts a vacancy at a constant ow cost c, and an active rm paired up with a worker gains an output ow p and incurs a labor cost w i p + p : In addition to the ow costs of posting vacancies, we follow Mortensen and Nagypál (2007) 1 in assuming that there is a one time hiring and training costs k (training costs for short) when a worker and a rm meet. We assume that this cost is transferable, and split between the two parties by the same type of generalized Nash bargaining as in the wage negotiations. As a result, a rm and a worker end up incurring the respective costs k f and k w to start an employment relationship. This cost captures in a simpli ed fashion the fact that rms incur hiring and trainings costs when they recruit new employees, and workers typically su er human capital losses when they undergo a spell of unemployment. Although most properties of our model do not depend on k being strictly positive, we believe that a successful numerical implementation of the model requires taking into account the full labor turnover costs. The search frictions in the labor market are characterized by a constant returns to scale matching technology: M (v; u) : The function M maps vacancies posted v and unemployment u onto the number of successful matches formed. Let be the vacancyunemployment ratio (v=u, also called market tightness) : The constant returns to scale of M implies that the rate at which workers nd jobs ( nding rate) is just a function of : f () = M (v; u) =u = M (; 1) : Likewise, the rate at which rms ll vacancies ( lling rate) satis es: q () = M (v; u) v = M 1; 1 = f () : (1) 1 The importance of training cost, or more generally turnover costs, for the dynamics of unemployment was earlier emphasized by Braun (2005), Nagypál (2005), Silva and Toledo (2005), and Yashiv (2005). 7

8 The function M is assumed continuously di erentiable, increasing in both arguments, and concave. Furthermore, it satis es the terminal conditions: M (1; 0) = M (0; 1) = 0, and M 1 (0; 1) = M 2 (1; 0) = 1. Therefore, workers nd it easier to nd jobs when vacancies are abundant relative to unemployment (in booms), while rms nd it easier to ll their vacancies when the reverse is true (in recessions). 2.2 Bellman Equations Workers may be in four possible states depending on whether they are employed or not and whether they are eligible for UI bene ts or not. Analogously, rms paired with a worker may be in two possible states depending on the worker s UI eligibility state. Contingent on productivity being p; let the values of being an employed worker and an unemployed worker, respectively, be Wp i and Up, i where superscript i denotes the worker s UI eligibility state: Similarly, let the values of a rm matched with a worker with UI eligibility state i be Jp. i Finally, when the economy experiences a productivity change (p! p 0 ), let the expression E p X i p0 denote the expected value of X (W; U; or J) conditional on p. Using this notation, the utility values Wp; i Up; i and Jp i for i = 0; 1 are recursively determined by the following Bellman equations. The value of an unemployed worker who does not collect UI is the present discounted value of the utility from leisure plus the expected gains from transitions to employment, which comes with an arrival rate f ( p ) ; or to a di erent productivity state, which comes with arrival rate. When a transition to employment happens, the worker incurs the training costs k w : ru 0 p = ` + f ( p ) (W 0 p U 0 p k w ) + (E p U 0 p 0 U 0 p ): (2) The analog equation for the value of an unemployed worker collecting UI includes the present discounted value of the utility from both leisure and UI bene ts and the expected gains or losses from transitions to employment, UI ineligibility, and a di erent produc- 8

9 tivity state. These transitions come at the arrival rates f ( p ) ; d; and ; respectively: ru 1 p = ` + b + f ( p ) (W 0 p U 1 p k w ) + d U 0 p U 1 p + (Ep U 1 p 0 U 1 p ): (3) The value of an employed worker ineligible for UI is the present discounted value of wages plus the expected gains or losses associated with exogenously losing the job, becoming eligible for UI, and experiencing a productivity change. The arrival rates of these events are s; g; and ; respectively: rw 0 p = w 0 p + s U 0 p W 0 p + g W 1 p W 0 p + (Ep W 0 p 0 W 0 p ); and (4) A worker eligible for UI can choose to quit the job to collect UI with probability instead of continuing with the match. The ow utilities attained with these two choices are respectively the rst and second arguments of the max operator in the following equation: rw 1 p = max (1 ) ru 0 p + ru 1 p ; w 1 p + s U 1 p W 1 p + (Ep W 1 p 0 W 1 p ) : (5) Upon quitting, the worker gets the expected utility of being unemployed, otherwise the worker gets the wage plus the expected capital gains or losses associated with losing the job exogenously or experiencing a transition to a di erent productivity. Because of free entry, the value of an unmatched rm is zero. The value of a rm employing a worker is the present discounted value of current pro ts plus the expected gains or losses associated with the worker becoming eligible for UI, the match exogenously dissolving, and productivity changing, which occur with arrival rates g; s; and. At any time, a rm can terminate the match, so the values of a matched rm cannot be negative. Given our assumptions, the value of a rm employing a worker ineligible for UI is always positive, 2 but the same cannot be assured if the worker is eligible for UI. Consequently, rj 0 p = p p w 0 p + g J 1 p J 0 p sj 0 p + (E p J 0 p 0 J 0 p ): (6) 2 As proven in Proposition 1, V 0 p > 0: Therefore, (11) implies J 0 p > 0 9

10 rj 1 p = max 0; p p w 1 p sj 1 p + (E p J 1 p 0 J 1 p ) : (7) Unmatched rms do not post vacancies if there are no expected gains in lling them, so p = 0 if J 0 p k f 0: Otherwise, unmatched rms post vacancies until the ow costs of posting a vacancy is equal to the expected gains of lling it, which occurs with an arrival rate q ( p ) ; so c = q ( p ) J 0 p k f : Using (1) and f( p ) 0; these two relations can be summarized as follows: c p = f ( p ) max 0; J 0 p k f (8) Since we abstract from the possibility of workers carrying UI eligibility earned from past employment to a new job, rms get the same value from matching with a worker who is collecting UI as the one who is not. Hence, it is consistent to assume that there is a single labor market where all workers and all rms interact. We leave the complexities derived from a dual labor market that di erentiates workers depending on if they are eligible for UI or not to future research. 2.3 Nash Bargaining The surplus of an employment match depends on the worker s entitlement to receive UI in case the match were dissolved. If the worker is not eligible for UI, the match surplus is de ned as: V 0 p = W 0 p U 0 p + J 0 p : (9) If the worker is eligible for UI, the match surplus depends on if the potential dissolution would be considered a quit or not by the UI agency. Because the agency imperfectly monitors why employment separations occur, we assume that if a match were to break down while bargaining, the worker would be able to collect UI with probability. This is the same probability of collecting UI after a voluntary quit because a worker who quits can be considered as one who cannot successfully negotiate a suitable pay raise. This assumption implies that the worker s opportunity cost of employment is (1 ) U 0 p +U 1 p : 10

11 Consequently, the match surplus when a rm bargains with a worker eligible for UI is: V 1 p = W 1 p (1 ) U 0 p U 1 p + J 1 p : (10) The generalized Nash solution to the bargaining problem maximizes the weighted product of the match surpluses of the two parties: J i p 1 (V i p J i p) ; where i takes values 1 or 0 depending on the UI eligibility state, and denotes the worker s bargaining power. The solution to this problem leads to the familiar sharing rule: J i p = (1 ) V i p ; for i = 0; 1: (11) Similarly, when a rm and a worker rst meet, the surpluses of both parties must subtract the training costs required for employment relationship to commence, so generalized Nash bargaining implies: J i p k f = (1 ) V i p k ; for i = 0; 1: (12) The combination of (11) and (12), together with k f + k w = k; results in the following split of the training costs: k f = (1 ) k; and k w = k: (13) 2.4 Equilibrium A recursive stochastic equilibrium is a set of eleven functions p ; w 0 p; w 1 p; U 0 p ; U 1 p ; W 0 p ; Wp 1 ; Jp 0 ; Jp 1 ; Vp 0 ; Vp 1 that satisfy the Bellman equations (2) to (7), the free entry condition (8), the match surplus de nitions (9) and (10), and the Nash bargaining solutions (11) to (13). This system of equations can be reduced to the following four functional equations (see the Appendix): c p = f ( p ) (1 ) max 0; V 0 p k ; (14) 11

12 ^U p = b + (E ^U p p 0 ^Up ) ; (15) r + d + f ( p ) ( s ^B p = max ^U ) p + (E p ^Bp 0 ^Bp ) ; r + s + g ^U p Vp 0 ; and (16) Vp 0 = p ` f ( p) Vp 0 k + g ^B p p + (E p Vp 0 V 0 0 r + s p ) : (17) Equation (14) is just the free entry condition (8) combined with the Nash bargaining rules (11) to (13). Equation (15) states that the value of UI eligibility for an unemployed worker, ^U p U 1 p U 0 p ; is equal to the expected present discounted value of the UI bene ts received by an eligible worker during a spell of unemployment. Equation (16) calculates the incremental value of achieving UI eligibility, which depends on if the match breaks down or not because of such eligibility. If the match survives UI eligibility ( rst term in 16), ^B p is the di erence between the expected present discounted values of the UI bene ts to be received upon an exogenous separation of the current match if the worker is eligible for UI (i = 1) or not (i = 0), which are equal to: Bp 1 = s ^U p + (E p Bp 1 0 B1 p) ; and (18) r + s Bp 0 = g B1 p Bp 0 + (Ep Bp 0 0 B0 p) : (19) r + s If UI eligibility kills the employment match (second term in 16), then ^B p is the expected value of UI eligibility for an unemployed worker minus the value of the match. Finally, equation (17) states that the value of the match between a rm and a worker ineligible for UI is the expected present discounted value of the bene ts resulting from the match. These bene ts include the labor productivity net of both the value of leisure and the workers expected value of nding a new job if the match breaks down, p ` f ( p ) V 0 p k, the net bene ts from the UI system, g ^B p p ; and the expected gains from a productivity change, (E p V 0 p 0 V 0 p ). Notice that the UI contribution p detracts from the value of the match exactly in the same way as the value of leisure does, but UI bene ts have exactly the opposite e ect. This implies that instead of reducing the value of a match, UI bene ts make the match more attractive at least before workers become eligible to collect them. 12

13 The equilibrium functions p ; ^Up ; ^Bp ; and V 0 p solve (14) to (17), and the remaining functions that de ne an equilibrium follow recursively from these four. The following proposition establishes the existence and some basic properties of an equilibrium. Proposition 1: An equilibrium exists and has the following properties: U 1 p > U 0 p and V 0 p > 0 for all p 2 P: Furthermore, if s= (r + s + g + ) ; then V 1 p > 0 for all p 2 P: (see the proof in the Appendix). As one would expect that an unemployed worker bene ts from being eligible for UI, and the match surplus is always positive if the worker is not eligible for UI. Also, if the probability of collecting UI is low when a worker eligible for UI quits a job, then the match surplus remains positive once eligibility is achieved. The following two propositions state additional properties of this equilibrium. Proposition 2: If workers are always denied bene ts after quitting a job voluntarily ( = 0) and the UI system is fully funded by UI contribution fees (each worker is charged the expected present discounted value of expected UI bene ts), then the level of UI bene ts, the duration of these bene ts, and the time it takes to become eligible for UI are irrelevant for the determination of output, vacancies, and unemployment. In particular, the introduction or elimination of a fully funded UI system with = 0 has no e ect on these variables. Proof: Since = 0; moral hazard quits never occur. Consequently, the expected present discounted value of UI contributions from a newly employed worker is: T p = p + (E p T p 0 T p ) : (20) r + s For the UI system to be fully funded, T p must be equal to B 0 p: Comparison of (19) and (20) implies that this equality holds if and only if p = g ^B p : So, (17) implies that V 0 p is 13

14 independent of b; d; and g: Therefore, neither p ; nor output, unemployment, or vacancies depends on these variables. Proposition 3: As long as there are no moral hazard quits, the equilibrium paths of vacancies and unemployment are independent of the probability of collecting UI after quitting a job voluntarily. Proof: Workers have no incentive to quit after they become eligible for UI if and only if the rst argument in the max operator in (16) does not fall short of the second one, and if this holds for all p 2 P; drops out from the system of equations (14) to (17), which determines p and so output, unemployment, and vacancies. Propositions 2 and 3 taken together provide a set of conditions that render a UI system irrelevant. Like other irrelevance results, such as Ricardian Equivalence, these propositions should be useful to pin point the economic e ects of a UI system as violations from their stated premises. In this vein, the e ects of a UI system have to be found in incorrectly pricing its insurance services, moral hazard, and risk aversion. More precisely, the adverse e ects of UI program on output and employment have to be found either in the way it is nanced, which may distort job creation, or in the rules for the provision of bene ts, which may engender strategic behavior such as quitting once eligibility is achieved or not searching while bene ts last. Also, with risk aversion, the bene ts of reducing income uncertainty with UI bene ts a ect the willingness to work and save in the ways that are beyond the scope of the present contribution. 3 Deterministic Equilibrium To obtain sharp results, this section follows Shimer (2005) and Mortensen and Nagypál (2007) and analyzes the special case where p is deterministic. As argued by Mortensen and Nagypál (2007), the comparative statics analysis of this deterministic model provides a good approximation for the dynamics of the stochastic model if productivity shocks are rare! 0; or they occur frequently but their changes are small. 14

15 As we will see, the predictions of how the economy reacts to shocks, such as a rise in productivity or an increased generosity of UI bene ts, depends crucially on the assumptions we make about the UI contribution fee : On one extreme, we can assume that is an endogenous variable that adjusts to maintain the UI system fully funded. On the other extreme, we can assume that is an exogenous parameter not a ected by the shocks considered. For this second assumption to be logically consistent in a general equilibrium context, we need to extend the model and assume that includes both UI contributions and general taxes, and that the government provides a public good, which yields separate utility. With this extension, when is kept constant while other parameters change, we are implicitly assuming that the government adjusts the provision of the public good endogenously to balance its budget. 3.1 Exogenous With the simpli cation that p is deterministic, the system of equations (14) to (17) that characterizes an equilibrium can be reduced to the crossing of the two schedules depicted in Figure 1. These schedules relate the value of a newly formed match V 0 with the vacancy-unemployment rate as follows. Schedule JC (job creation) represents the free entry condition (14). Its upward sloping shape captures that rms respond to a rise in the expected pro ts associated with a rise in V 0 by posting more vacancies until the lling rate becomes su ciently low so that the value of posting a vacancy falls back to zero. Schedule MV (match value) is the representation of the mapping from to V 0 implied by the remaining equilibrium equations (15) to (17). Using (14), the absence of productivity shocks and the equilibrium properties that V 0 > 0, these equations simplify into: ^U = b r + d + f () ; (21) ( s ^B = max ^U ) r + s + g ; ^U V 0 ; and (22) V 0 = p ` (1 ) 1 c + g ^B : (23) r + s 15

16 Equation (23) implies that there are two reasons why the value of a match V 0 falls with as represented in Figure 1. First, as indicated by (1 ) 1 c; workers nd jobs easier if there are more vacancies posted, which pulls up the wage due to the improved bargaining power and lowers the match surplus. Second, as captured by g ^B; the expected present discounted value of the UI bene ts received by an eligible worker in (21) falls with the job nding rate and so with : As a result, the value of the jobs needed to gain this eligibility falls as well. Figure 1 is useful to analyze the qualitative implications of the model. For example, an increase in training costs k shifts the JC schedule up ( rms post less vacancies), so it leads to a rise in V 0 and a fall in : In contrast, an increase in labor productivity or a fall in the value of leisure shift the MV schedule up (matches become more valuable), so both V 0 and increase. A more generous provision of UI bene ts (a rise in b or g; or a fall in d) also shifts the MV schedule up because the matches that would make workers eligible for UI bene ts become more valuable. Meanwhile, a more expensive UI contribution (a rise in ) has the opposite e ect on the value of matches because it is equivalent to an increase in the value of leisure `. Consequently, in contrast with models where workers do not need to accumulate the employment time to become eligible for UI, a more generous UI system and a more expensive one have competing e ects on the vacancy-unemployment ratio : Therefore, our model is able to reconcile the sharp response of V 0 ; ; v; and u to productivity improvements with a mild response of them to changes in b and if these changes tend to happen together. Depending on whether rms post vacancies in equilibrium or not, and whether workers eligible for UI quit their jobs or not, we can distinguish four possible types of equilibria: Normal: V 0 > k V 1 0; Strategic: V 0 > k V 1 0; Phase-out: V 0 k V 1 0; Autarky: V 0 k V 1 0; where V 1 is the value of continuing the match once UI eligibility is achieved: V 1 = 16

17 V 0 + [s= (r + s + g) ] ^U (Since dissolving the match is an option, the value of the match is V 1 = max 0; V 1 ). Vacancies are posted in equilibrium if and only if the value of a newly formed match exceeds the training costs (V 0 > k) ; and employed workers have no incentive to strategically quit a job if the value of continuing the match is not exceeded by the expected UI bene ts received by quitting ( V 1 0): In the normal equilibrium, both of these inequalities hold, so new jobs are created and matches survive when workers become eligible for UI. In the strategic equilibrium, the rst inequality holds but not the second. So, new employment matches are formed, but they break down as soon as workers become eligible for UI. Finally, in the phase-out equilibrium and the autarky equilibrium, no new jobs are created. If the value of initial employment is positive, the worker- rm pair maintains until an exogenous separation comes in the phase-out equilibrium, while workers quit as soon as they become eligible for UI in the autarky equilibrium. For V 0 to exceed k; the MV schedule in Figure 1 must cross the JC schedule above k: Consequently, the parameters that shift the MV schedule up and the JC schedule down must be su ciently large relative to those that have the opposite e ects. In particular, we need that the match is su ciently productive and/or UI bene ts su ciently generous relative to the cost of posting a vacancy, the value of leisure, UI contributions-taxes, and training costs. For a match to survive once the worker is eligible for UI ( V 1 0), the probability of collecting bene ts or the present value of the bene ts ([s= (r + s + g) ] ^U) must be su ciently low relative to V 0 : Figure 2 depicts how k and interact in the determination of the various types of equilibria (see the Appendix for its construction). As long as V 1 0 ( is su ciently low), has no e ect on V 0 ; so the V 0 = k line is horizontal at the value k in which this equality is satis ed. Once is su ciently high for workers to quit upon receiving UI eligibility ( V 1 0); an increase in makes an employment match more valuable, so the V 0 = k line is upward sloping. As long as V 0 k (k is su ciently large); k has no local e ect on the value of a match, so the line ( V 1 = 0) is vertical at the probability in which this equality is satis ed. However, once k is su ciently low for new jobs to be created (V 0 > k) ; a reduction in k (downward shift of JC line in Figure 1) reduces V 0 17

18 and increases : As more vacancies are created, the job nding rate f () goes up, which reduces the value of UI eligibility ^U and so ^B. Consequently, the reduction in k brings down both the value of continuing a match (V 0 + ^B) and the value of quitting ( ^U) once a worker is eligible for UI, so it has an ambiguous e ect on the value of needed to maintain the equality V 1 = 0: This implies an ambiguous slope for the line V 1 = 0 in the region where V 0 > k. 3 Since V 0 is guaranteed to be positive and nite, and moral hazard quits are ruled out if < s=(r + s + g); on regions in Figure 2 where the normal and the phase-out equilibria exist are never empty. However, depending on how productive matches are and how generous the UI system is, moral hazard quits may not happen even if = 1; in which case there is no strategic or autarky equilibria for all admissible values of : Increasing the net productivity of a match (p ` ) or the generosity of UI bene ts (a rise in b or g; or a fall in d) makes a newly formed match more valuable, so it shifts up the V 0 = k line in Figure 2. Likewise, even if the worker is eligible for UI, the value of the match increases with net productivity, so a rise in (p ` ) shifts the V 1 = 0 line to the right. However, the generosity of UI bene ts has an ambiguous e ect on the location of this line, because a more generous UI system raises both the value of continuing the match and the value of quitting. That is, paradoxically increasing the generosity of the UI system may prevent quits in some regions of the parameter space. In the region where V 0 k, the second e ect is always dominant, so if b or g rise or d falls, V 1 goes down and the line V 1 = 0 line shifts left. However, we cannot be certain that a shift in the same direction occurs in the region where V 0 > k: To study the quantitative predictions of the model, we can apply the standard comparative statics methodology to the equilibrium system of equations (14) and (21) to (23). Of particular interest is the elasticity of the nding rate with respect to labour productivity because it gives a good indication of the amplitude of the labor market cycles generated by productivity shocks (see Mortensen and Nagypál, 2007). As long as > 0, 3 As depicted in Figure 2, in a neighborhood of k; the slope of this frontier must be negative (see Appendix). 18

19 this elasticity is the following (see the Appendix): 8 df p >< dp f = >: p p p p z z " (1 ) (r + s) + f (r + s + f) + " (1 ) (r + s + g) + f (r + s + g + f) f g ^B # 1 if V 1 > 0; and r + d + f p z + f g ^U r + d + f p z # 1 if V 1 < 0: (24) where is the elasticity of f with respect to ; and 8 >< ` + + (r + s) k g ^B if V 1 > 0; and z = >: ` + + (r + s + g) k g ^U if V 1 < 0: (25) In the absence of training costs and UI, the second term inside the square brackets in (24) drops and z = `; so the nding rate responds to changes in productivity as derived in Mortensen and Nagypál (2007). As Shimer (2005) pointed out, for reasonable parameter values and a low value of leisure, this response is too small to generate the pronounced cycles in the United States labor market. A high value of leisure, by making the pro ts margin p z small, leads to a su ciently large elasticity of f with respect to p to rationalize the observed responses over the business cycle. For example, Hagedorn and Manovskii (2007) found that when ` equals 0:97; model is able to generate the variance of observed in the United States business cycles. As long as the equilibrium remains normal or strategic, (25) implies that the pro t margin falls with and k; so a small pro t margin may be compatible with a relatively small value of ` if UI contributions, taxes, and training costs are large. However, the e ect of UI bene ts on the pro t margin works in the opposite directions. Positive UI bene ts also add the second terms inside the square in (24), which further decreases the elasticity of f with respect to p: Since p; `; and only enter the equilibrium system of equations (14) and (21) to (23) in the determination of the pro ts margin, an increase in ` or a ects f in the same way as a reduction in p of the same magnitude does. Similarly, UI bene ts enter the determination of an equilibrium through the term g ^B: Therefore, di erentiation of (21) to (23) implies: 19

20 8 df >< db = >: df g s if V 1 > 0; and dp r + d + f r + s + g df g dp r + d + f if V 1 < 0: As pointed out in the qualitative analysis of Figure 1, a rise in b unambiguously increases and so the job nding rate. As long as the increase in b does not trigger moral hazard quits, this implies that, paradoxically, a more generous UI system reduces unemployment. However, a move towards a more generous UI system may trigger a shift from the normal to the strategic equilibrium in which case the level of steady state unemployment will experience a discontinuous jump. Indeed, for the stock of unemployment to be constant, the ows into unemployment must equal the ows out of unemployment. Denoting u ss as the steady state unemployment, we have: (26) (1 u ss ) s = u ss f in the normal equilibrium, and (1 u ss ) (s + g) = u ss f in the strategic equilibrium. (27) Solving for u ss from these equations yields: u ss = s s + f u ss = s + g s + g + f in the normal equilibrium, and in the strategic equilibrium. (28) Consequently, the e ect of UI generosity on unemployment is non-monotonic. Even though a rise in b reduces u ss through increases in f; it may also trigger moral hazard quits in which case the e ective separation rate is s + g instead of s: The discontinuous jump in u ss predicted in this model is an artifact of workers being homogenous. With heterogeneity, a rise in b could increase or reduce u ss depending on how many moral hazard quits it triggers. 20

21 3.2 Fully Funded UI System If the UI system is fully funded, then using an argument analogous to the one in the proof of Proposition 2, the value of has to adjust to satisfy: 8 >< g ^B if V 1 0; and = >: g ^U if V 1 0: (29) Therefore, the value of a new match simpli es into 8 >< V 0 = >: p ` (1 ) 1 c r + s p ` (1 ) 1 c r + s + g if V 1 0; and if V 1 0: (30) As depicted in Figure 3, the endogenous adjustment of to maintain the UI system fully funded induces two di erent MV schedules depending on whether moral hazard quits occur (MV N ) or not (MV S ). The schedule MV N lies above MV S because moral hazard quits are costly to the UI system, so high UI contributions need to be imposed. Consequently, both the equilibrium value of a new match and the vacancy-unemployment ratio are higher in a normal equilibrium relative to their counterparts in a strategic equilibrium with the same parameter values: V 0N ; N > V 0S ; S : Using (21) and (22), the conditions for these equilibria to be consistent with the incentive to quit or not are: s b V 0N r + d + f N implies no incentive to quit, and(31) r + s + g s b V 0S r + d + f S implies no incentive to continue. (32) r + s + g Notice that b; ; and d have no e ect on neither V 0N ; N nor V 0S ; S ; and g has no e ect on V 0N ; N and is inversely related to V 0S and S : Therefore, if the UI system is very generous (b; g; and are high, and d is low), then (32) is satis ed and (31) is violated, so moral hazard quits occur. On the other extreme, if the UI system is very stingy (b; g; and are low, and d is high), then (31) is satis ed and (32) is violated, so 21

22 moral hazard quits do not occur. Finally, since V 0N ; N > V 0S ; S ; there is a generic set of intermediate UI systems such that both (32) and (31) are satis ed. In which case, two di erent equilibria coexist. In one of them, workers quit once they become eligible for UI, and in the other they do not. The intuition for the generic multiplicity of equilibria when endogenously adjust to maintain the UI system fully nanced is the following. In the "good" equilibrium, UI contributions are relatively low because workers eligible for UI do not quit, and workers have no incentive of quitting because the UI contributions they have to pay if they remain employed are low. Vice versa, in the "bad" equilibrium, UI contributions need to be large to nance the expensive UI payments since all workers quit upon earning UI eligibility, and these workers have incentives to quit because if they remain employed, they are burdened with large UI contributions. Figure 4 illustrates the regions of the coexistence of the various types of equilibria for some intermediate values of : For low values of k; new jobs are created, so the normal equilibrium may coexist with the strategic equilibrium. For high values of k; no new jobs are created, so the phase-out equilibrium may coexist with the autarky equilibrium. Finally, for intermediate values of k; the normal equilibrium may coexist with the autarky equilibrium. In this case, the high UI contributions need to nance the expensive UI system not only give workers incentives to quit once they are eligible for UI, but they also shut down the creation of new jobs. 4 Labor Cycles in Canada and the United States This section calibrates the model to data from Canada and the United States allowing for the value of leisure in these two countries to be as high as needed to generate a realistic large volatility in the unemployment-vacancy ratio. In particular, we examine if the similar labor market cycles experienced in the two countries can be generated with reasonably similar values of leisure. This proves to be an insurmountable challenge for the standard version of the Mortensen-Pissarides model where entitlement to UI does not 22

23 need to be earned (see Zhang, 2008). The reason for this di culty is that Canada has both higher taxes and UI bene ts than the United States, which is inconsistent in those models with the similar amplitude of the cycles experienced by unemployment and vacancies. In the present model, taxes and UI bene ts a ect the opportunity cost of employment in opposite directions, so there is hope that this challenge can be met. This section also examines if the calibrated values of leisure are neither too high for unemployment not to be a major social concern nor too low to make the UI rules trying to prevent moral hazard behavior nonsensical. Finally, we enquire about the response of unemployment to increases in UI bene ts and taxes. The numerical simulations in the calibration use a discrete time version of the model analyzed in Sections 2 and 3 with the following specializations. The matching function is assumed to be Cobb-Douglas: M(v; u) = u 1 v ; where is the elasticity of the nding rate with respect to the vacancy-unemployment ratio: f () = : Also, consistent with Shimer (2005), labor productivity is assumed to follow a stochastic process that satis es: p = ` + + e y (p ` ); where p is normalized to one, and y is a zero mean random variable that follows an eleven-state symmetric Markov process in which transitions only occur between contiguous states. As detailed in the Appendix, the transition matrix governing this process is fully determined by two parameters: the step size of a transition, ; and the probability that a transition occurs,. The model period in the simulations is chosen to be one month, so the real interest rate is set to the conventional monthly rate of 0:4 percent. Even if the model period is one month, consecutive periods are aggregated to construct quarterly series to match empirical moments at that frequency. The calibration targets, summarized in Table 1, aim to replicate the main rates, the labor market ows and, in a stylized way, the key features of the taxation and UI systems in the two countries. The data sources and methodological details in calculating these targets can be found in the Appendix. 23

24 Table 1 Calibration Targets U.S. Canada Monthly real interest rate (r) 0:004 0:004 Average monthly nding rate (f) 0:452 0:309 Average monthly unemployment rate (u) 0:0567 0:0778 Elasticity of nding rate with respect to () 0:54 0:54 Average vacancy-unemployment ratio 1 1 Standard deviation of (quarterly in logs) 0:151=0:382 0:191=0:367 Standard deviation of of labor productivity (quarterly in logs) 0:020 0:021 Autocorrelation of labor productivity (quarterly in logs) 0:878 0:876 Average weeks of employment needed for UI eligibility (1=g) Average weeks before UI bene ts expire (1=d) Average actual UI bene ts replacement rate bu 1 =wu 0:111 0:265 Average tax rate inclusive of UI contributions () 0:30 0:35 Ratio of training costs to quarterly wage rate (k=w) 0:55=0 0:37=0 Standard deviation of real wage w (quarterly in logs) free=0:012 free=0:016 From the labor market, the calibrations aim to replicate the standard deviation and autocorrelation of detrended labor productivity, the average monthly nding and unemployment rates, the standard deviation of the vacancy-unemployment ratio ; and the elasticity of the nding rate with respect to the market tightness. Detrended labor productivity and the nding rates were calculated using the same methodologies as in Shimer (2005). The average unemployment rates are directly calculated using standard data from both countries over the sample periods for the United States, and for Canada. The average vacancy-unemployment ratio is normalized to be one, which implicitly de nes the units in which vacancies are measured and sets the value of to be the average monthly nding rate. The standard deviation of is used as the gauge of the amplitude of the cyclical uctuations in the labor market. In the baseline calibration, we follow Mortensen and Nagypál (2007) and target the standard deviation of conditional on p; recognizing in this way that productivity shocks are not the only source of cyclical variations. However, to check how much our results depend on this choice, we also report calibrations using the unconditional standard deviation of as the target. Finally, the elasticity of the nding rate with respect to the market tightness is estimated using the method proposed by Mortensen and Nagypál (2007), which uses the 24

25 law of motion of unemployment at the steady state and sets = 0:54 in both countries (see Appendix for details). From the UI programs, the calibrations aim to be consistent with the average time it takes for a worker to gain UI eligibility, the average duration of UI bene ts, and the average actual replacement rates of UI bene ts in the two countries. In the United States, UI eligibility takes around 20 weeks of work and the maximum duration of bene ts is around 24 weeks 4. In Canada, both the time needed for eligibility and the maximum duration of bene ts have changed over time and currently depend on the unemployment rate in the region of residence. The targets used in the calibration, 15 weeks to gain eligibility and 33 weeks for the maximum duration of bene ts, are representative gures over the sample period 56. The average actual replacement rate of UI bene ts is de ned as the ratio of the average weekly UI bene ts paid to unemployed workers over the average weekly insurable earnings paid to employed workers. As explained in the Appendix, these rates are obtained as the product of two ratios. The rst ratio is the average weekly UI bene ts paid to UI recipients over the average weekly insurable earnings paid to employed workers (b=w). The second ratio is the fraction of unemployed workers receiving UI bene ts (u 1 =u). Finally, the parameter is interpreted not just as UI contributions but also as a general tax, so the government is using a large fraction of to nance a public 4 Card and Riddell (1992) documents that in most states in 1989, UI eligibility requires 20 weeks of work, or the earnings equivalent of 20 weeks of full-time work at the minimum wage, and the maximum duration of bene ts lasted around 24 weeks. Similarly, Osberg and Phipps (1995) compares the UI eligibility requirements across states and nds that Texas (relatively less generous state) and New York (relatively more generous state) both set 20 weeks as the minimum employment weeks to qualify in As to the entitlement UI weeks in Canada, under the UI Act of 1971, regular UI eligibility required a minimum of 8 employment weeks during the base year. In 1977, the minimum employment weeks was replaced by variable entrance requirement (VER) and increased to weeks in 1977, then to weeks in E ective in 1997, the VER based on employment weeks was replaced by an entrance requirement based on hours of work. The minimum hours for regular UI bene ts ranged from hours. We link these two VERs by converting hours of work to full-work weeks. For example, 420 hours is equivalent to 10.5 weeks of full-time work. 6 With respect to the maximum duration of bene ts, as reported in Table 4 in EI Reform and Multiple Job-Holding - November 2001 from Human Resource and Social Development Canada, it was 32:8 weeks in 1995, and 32:9 weeks in The legislations regarding the UI duration are rather complicated. For example, Canada has a vestage bene ts structure. The UI duration (after 1989) depends on the previous weeks of work and the prevailing unemployment rate in the region of residence. These complexities make it impossible to calculate an average over the sample period. 25